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# Re: Programming is the Engineering Discipline of the Science that is Mathematics

From: vc <boston103_at_hotmail.com>
Date: 11 Jun 2006 18:23:56 -0700

> vc wrote:
> >
> >>Are you seriously suggesting that true and false are trivial and
> >>uninteresting? Should we all pack up and go home?
> >
> > I am suggesting that a meaningful PT statement reduction to a
> > propositional logic statement is trivial and uninteresting.
>
> Then why did you bring it up in the first place?

If you read the whole thread carefully, you'll realize that it was the OP who brought the 'reduction' issue, not me:

<OP>
"
This is why PT is a /generalization/ of logic. It reduces to logic when applied to truth-valued statements. Just as gamma reduces to factorial for natural arguments. (Again no quibbles about offset by 1 etc).
"

My very first message just modestly reminded that probability is not truth functional, that was all.

>
>
> Also,
> > there is a problem with the conditional probability not being equal
> > the probability of the conditional (see Lewis's result) which makes
> > conditional probabily untranslatable to modus ponens in principle.
>
> Modus ponens requires a conditional probability of 1 and says nothing
> about situations with other conditional probabilities which is why the
> conditional "If A then B" says nothing about B when A is false.
>
> For modus ponens, one must start with the premise "If A then B", which
> is synonymous with P(B|A) = 1. To have a sound argument, one must not
> only have a valid argument, but the premise "If A then B" must be true.
>
> It think your argument might cause more problems for modus tollens than
> for modus ponens.

The proponents of the probabilistic logics hoped that P(A|B) = P(B->A).  Lewis showed that the conditional probability cannot be the probability of implication (the truth functional conditional) thus making truth-functionality impossible though the conditional probbaility either.

>
>
> > Going in the opposite direction, generalization, PT is not truth
> > functional , that is the probability of a compound statement is not
> > determined solely by its components probabilities (see my trivial
> > puzzle).
>
> I fail to see what an indeterminate problem demonstrates. If I give you
> the lengths of two sides of a triangle and ask you for the length of the
> third side, you cannot answer unless you know the angle where the given
> sides meet.

The problem merely demonstrates that PT does not possess truth functionality in the same way the propositional logic does. In order to derive the compound statement probability additional information must be taken into account while with PL the compound statement truth depends only on truth values of its sub-propositions.

>
> Such a challenge would neither disprove pythagoras nor disprove the law
> of cosines. Neither would it disprove that the law of cosines
> generalizes the pythagorean theorem.

That's an irrelevant remark. See above.

>
>
> Also, importantly, it appears impossible to find an axiom
> > system for any known probabilistic logic that would be sound and
> > complete (except some special cases). Obviously, lack of such axiom
> > system makes a formal derivation (a hallmark of any logic) impossible.
>
> That precludes it from deductive logic but not from inductive logic.

That very well may be the case depending on what you mean by "inductive logic", but the original statement appears to be that PT somehow subsumes/"generalizes" propositional logic, or any other truth-functional/deductive logic, which is a bizzare statement indeed. If some other, non-truth functional logic was implied, then the context should have been clearly stated, the usual assumption being that the '[default]logic' = 'propositional logic/predicate calculus'.

>
>
> > Apparently, despite obvious similarities and profound connections,
> > both had better be used what they are best at and attempts to merge
> > them do not seem very productive (see abundant literature on
> > probabilistic logics).
>
> Are you suggesting that they need merging?

You are confused, the OP does (see his talk about "generalizing"), not me.

>Classical mechanics is a
> special case of relativity. Do they need merging? The pythagorean
> theorem is a special case of the cosine law. Do they need merging?
> Deductive logic is a special case of inductive logic. Do they need merging?
>

You are preachig to the choir.

> Each of the special cases has properties that do not generalize and each
> of the generalizations considers factors irrelevant to the special case.
>
> The real question is whether inductive logic is appropriate for data
> management, and I do not think it is.

Neither do I.

>
> >>>What Jaynes did in his derivation of the sum/product rules has got
> >>>nothing to do with your mindless playing with formulas. See the
> >>>argument from authority in my previous messages.
> >>
> >
> > The argument from authority was a quote from Jaynes' book , not mine.
>
> The source of the quote is irrelevant because the flaw itself was a lack
> of relevance. Keith never relied on the meaning of any meaningless
> value. The fact that the value of x might be indeterminate is
> unimportant to the conclusion that x times zero is zero because the
> conclusion holds for all x.

He relied on the non-existing value, not an existing but unknown/indeterminate value. There is no x*zero simply because x does not exist. Again, the Bayesian conditional probability *exists* only if the respective conditioning proposition is true. The very derivation of the product and sum rules from the Cox postulates relies substantially on B being true in P(A|B). For a rigorous exposition see for example Aczel's Lectures on Functional Equations.

>
>
> >>>>that should have been P(p1|p2) != P(p1).
> >>>
> >>>That's assuming that P(p1|p2) even makes sense. More general
> >>>formulation of such independence is just P(p1 and p2) = P(p1)* P(p2).
> >>
> >>The formulation is neither more general nor less general. It is, in
> >>fact, a simple substitution of the equation describing independence:
> >>
> >>(1) P(p1|p2) = P(p1)
> >
> > Again, the substitution is possible only when P(p2) > 0. (See the
> > Jaynes book).
>
> Okay, the formula for independence only applies when P(p2) > 0. And how,
> exactly, does that support your assertion that one formulation is more
> or less general than the other?

Independence can be (and is) defined as P(A and B) = P(A)*P(B). It works with whatever values of P(A) and P(B) as opposed to the conditional probability formulation.

>
>
> >>into the formula for conditional probability:
> >>
> >>(2) P(p1 and p2) = P(p1|p2)*P(p2)
> >>
> >>Substitute (1) into (2) gives P(p1 and p2) = P(p1)*P(p2)
Received on Sun Jun 11 2006 - 20:23:56 CDT

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